Forward rates, Black's formula caplet by caplet, the split between intrinsic and time value, and the expiry mistake that overstates the premium by a fifth.
An interest rate cap is priced as a strip of caplets, each one a call option on a single forward rate valued with Black's model and discounted to its payment date. On an illustrative 50,000,000 four-year cap struck at 4.50 per cent, with forwards between 4.40 and 4.75 per cent and a 22 per cent volatility, the three caplets are worth 157,491, 283,420 and 349,152: a premium of 790,064, or 1.58 per cent of notional, about 59.8 basis points a year.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A borrower with a floating-rate loan of 50,000,000 resetting annually wants protection above 4.50 per cent for four years. The first year's rate is fixed today, so it carries no optionality: the cap has three live caplets, on the rates for years two, three and four. All rates and the volatility are illustrative; one curve is used to project and to discount so that the arithmetic stays visible.
| Input | Value |
|---|---|
| Notional | 50,000,000 |
| Strike | 4.50% |
| Black volatility, flat | 22% |
| Zero rates, years 1 to 4 (annual compounding) | 4.00%, 4.20%, 4.35%, 4.45% |
| Discount factors, years 1 to 4 | 0.961538, 0.921010, 0.880081, 0.840168 |
| Accrual fraction per period | 1.0 |
Each forward comes from two adjacent discount factors: F = DF(start) / DF(end) − 1. The year-one rate is 4.0000 per cent, already fixed and below the strike. The three forwards that matter are 4.4004 per cent for year two, 4.6506 per cent for year three and 4.7506 per cent for year four. Two of the three are already above the strike on the forward curve, which is why this cap is not cheap.
Caplet = N × τ × DF(payment) × [F × N(d1) − K × N(d2)]
d1 = [ln(F / K) + ½σ²T] / (σ√T), d2 = d1 − σ√T, T = time to the fixing date
Excel: =N*tau*DF*(F*NORM.S.DIST(d1,TRUE)-K*NORM.S.DIST(d2,TRUE)) with d1=(LN(F/K)+0.5*sig^2*T)/(sig*SQRT(T))
Take the year-two caplet. It fixes in one year, so T is 1 and σ√T is 0.2200. ln(4.4004 / 4.50) is −0.02239 and ½σ²T is 0.02420, so d1 is 0.0082 and d2 is −0.2118. N(d1) is 0.5033 and N(d2) 0.4162. The bracket is 2.21467 per cent less 1.87268 per cent, or 0.34200 per cent of notional, paid at the end of year two. Times 50,000,000 and the year-two discount factor of 0.921010, the caplet is worth 157,491.
| Period | Forward | T to fixing | d1 | d2 | Value | Intrinsic on forward | Time value |
|---|---|---|---|---|---|---|---|
| Year 2 | 4.4004% | 1 | 0.0082 | −0.2118 | 157,491 | 0 | 157,491 |
| Year 3 | 4.6506% | 2 | 0.2614 | −0.0497 | 283,420 | 66,291 | 217,129 |
| Year 4 | 4.7506% | 3 | 0.3327 | −0.0483 | 349,152 | 105,263 | 243,890 |
| Cap | 790,064 | 171,554 | 618,510 |
The cap costs 790,064 upfront, 1.5801 per cent of notional. Spread over the three protected years, using the annuity of the year two to four discount factors (2.641260), that is 59.8 basis points a year: the borrower is paying roughly 0.60 per cent a year to be sure its base rate never costs more than 4.50 per cent.
The split between intrinsic and time value is the useful part. Only 171,554 of the premium is the payoff the cap would deliver if the forward curve were realised exactly. The other 618,510, more than three quarters, is the price of volatility: the chance that rates end higher than the curve says. The later caplets carry more of it because they fix later and the rate has longer to move, which is why σ√T rises from 0.2200 to 0.3811.
Vega is large. One more point of volatility, 23 per cent instead of 22, adds 32,280 to the premium. A quote that moves two vol points between the term sheet and execution changes the cost of the cap by about 65,000 without the rate curve moving at all.
| Strike | Vol 18% | Vol 22% | Vol 26% |
|---|---|---|---|
| 4.00% | 1,037,113 (78.5 bp) | 1,145,232 (86.7 bp) | 1,257,171 (95.2 bp) |
| 4.50% | 660,830 (50.0 bp) | 790,064 (59.8 bp) | 919,041 (69.6 bp) |
| 5.00% | 405,998 (30.7 bp) | 534,913 (40.5 bp) | 665,674 (50.4 bp) |
| 5.50% | 245,286 (18.6 bp) | 359,584 (27.2 bp) | 481,176 (36.4 bp) |
Moving the strike up 50 basis points saves about a third of the premium; moving it up a full point saves more than half. The trade-off a treasurer is actually making is between the strike and the volatility the market charges, and at higher strikes the volatility is a larger share of what is paid: at 5.50 per cent every caplet is out of the money on the forward curve, and the whole 359,584 is time value.
A cap is a sum of caplets, each a Black call on a forward rate, expiring at its fixing and discounted from its payment date. Here that is 790,064 for three years of protection at 4.50 per cent, of which only 171,554 is the curve and the rest is volatility. The same formula, with a put instead of a call, prices the floor, and a cap bought with a floor sold is the collar: whether a zero-cost collar is really zero cost runs Black-76 on exactly that structure, and does an interest rate cap protect your DSCR asks where to put the strike. Black-76 is written out in spreadsheet cells in the free workbook for the book's collar, so the strikes can be re-struck and the premiums watched.
Divide the upfront premium by the notional times the sum of the discount factors for the protected periods. Here 790,064 divided by 50,000,000 times 2.641260 gives 59.8 basis points a year. That is the running cost a borrower can add to its margin to compare a cap with a swap or with staying unhedged.
Each caplet is a call option, and an option's value rises with the uncertainty of the underlying. On this cap, 618,510 of the 790,064 premium is time value rather than intrinsic value, and one extra point of Black volatility adds 32,280. Volatility moves the price even when the forward curve does not move at all.
A caplet is a call on the interest rate: it pays when the rate fixes above the strike. A floorlet is a put on the rate. A caplet is also equivalent to a put on a zero-coupon bond, which is how some systems price it. On the year-two caplet here the call pays 0.34200 per cent of notional in expectation, worth 157,491 today.
This article is one calculation from Derivatives. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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